for \(n\in {N}\)  and \(P_0(x)=1\) , and we take \(P_n(1)=1\) , \(P_1(x)=\frac{1}{a_0}(x-b_0)\) 
with \(a_n>0\) , \(b_n\in {R}\) , and \(c_{n+1}>0\)  for all \(n\in {N}_0\) . The existence of the orthogonality measure is due
to Favard's theorem [1]} and applying it to the
relation (REF ) results in \(a_n+b_n+c_n=1\) 
and \(a_0+b_0=1\) . The identity map defines
an involution to these hypergroups and their Haar weights are given
by \(h(0)=1\)  and \(h(n)=\left(\int _{R}P_n^2(x)d\pi (x)\right)^{-\frac{1}{2}}\) 
\((n\ge 1)\)  [2]}. We consider the \(\alpha \) -amenability
of following polynomial hypergroups.
